Delay equalization and how it is done

Delay (phase) equalization is the process of making the total group delay of a filter or channel nearly constant over the passband, by cascading it with all-pass networks.

Filters with a sharp magnitude response (Chebyshev, elliptic, high-order Butterworth) have a delay that peaks near the band edge, which distorts pulses and data signals. An all-pass section has $|T| = 1$, so it does not change the magnitude response, but it adds its own delay. The total delay is the sum:

$$\tau_{total}(\omega) = \tau_{filter}(\omega) + \tau_{AP}(\omega)$$

How it is done:

1. Plot or compute the delay $\tau_F(\omega)$ of the filter over the passband.
2. Choose first- and/or second-order all-pass sections whose delay is large where $\tau_F$ is small and small where $\tau_F$ is large (a second-order section gives a delay peak near its $\omega_o$, with height set by $Q$).
3. Adjust $a$, $\omega_o$ and $Q$ (and add sections) until $\tau_F + \tau_{AP}$ is flat within the allowed tolerance.
4. Realize the all-pass sections with op-amp circuits and cascade them with the filter.

 tau      filter        equalizer        total
  |        .            .     .
  |      .' `.          `. .'         .------.
  | ...-'     `      ->   `     ->   '        `
  +----------- w      +------ w      +---------- w

First-order op-amp all-pass: two equal resistors $R_1$ in the inverting path and feedback, and an RC network at the + input, give $T(s) = \dfrac{1 - sRC}{1 + sRC}$.

          R1        R1
 Vi o--+-/\/\-+---/\/\---+
       |      |  |\      |
       |      +--|-\     |
       R         |  >----+--o Vo
       |      +--|+/
       +------+  |/
              C
              |
             GND

Applications: data modems, telephone lines, video and pulse transmission, where waveform shape must be preserved.